Volume of 3D Figures

Grade 8 · Math · Study Guide

Learn how to calculate the volume of cylinders, cones, and spheres using their formulas.

Volume measures the amount of three-dimensional space inside a solid figure. It is measured in cubic units, such as cm³ or in³.

A rectangular prism has volume V = length × width × height (V = lwh). A cube is a special prism where all sides are equal, so V = s³.

A cylinder has two circular bases. Its volume is V = πr²h, where r is the radius of the base and h is the height.

A cone has one circular base and tapers to a point. A cone holds exactly one-third the volume of a cylinder with the same base and height: V = (1/3)πr²h.

A sphere is a perfectly round solid. Its volume is V = (4/3)πr³, where r is the radius.

To solve volume problems, always identify the shape first, pick the correct formula, plug in the given measurements, and use 3.14 or the π key for calculations unless told otherwise. Pay close attention to whether you are given a radius or a diameter — remember that the radius is half the diameter.

Key terms

Volume
The amount of 3D space inside a solid, measured in cubic units
Cylinder
A solid with two parallel circular bases; V = πr²h
Cone
A solid with one circular base tapering to a point; V = (1/3)πr²h
Sphere
A perfectly round solid; V = (4/3)πr³
Radius
The distance from the center of a circle to its edge; half the diameter
Pi (π)
The ratio of a circle's circumference to its diameter; approximately 3.14159

Flashcards

Volume of a rectangular prism
V = l × w × h
Volume of a cube
V = s³ where s is the side length
Volume of a cylinder
V = πr²h
Volume of a cone
V = (1/3)πr²h
Volume of a sphere
V = (4/3)πr³
Radius vs. diameter
Radius = diameter ÷ 2; always use radius in volume formulas
Cubic units
The unit for volume; e.g., cm³, m³, in³
Cone vs. cylinder
Same base and height: the cone's volume is exactly 1/3 of the cylinder's

Questions

  1. What is the volume of a cylinder with radius 3 cm and height 10 cm? (Use π ≈ 3.14)

    1. 94.2 cm³
    2. 282.6 cm³
    3. 188.4 cm³
    4. 565.2 cm³
    Show answer

    Answer: 282.6 cm³

    V = πr²h = 3.14 × 3² × 10 = 3.14 × 9 × 10 = 282.6 cm³.

  2. The formula for the volume of a cone is V = (1/3) × π × r² × _____.

    Show answer

    Answer: h

    The cone formula is V = (1/3)πr²h, where h represents the height.

  3. A cone has exactly one-third the volume of a cylinder with the same radius and height.

    Show answer

    Answer: True

    The cone formula V = (1/3)πr²h is exactly one-third of the cylinder formula V = πr²h.

  4. What is the volume of a sphere with radius 3 cm? (Use π ≈ 3.14)

    1. 37.68 cm³
    2. 113.04 cm³
    3. 150.72 cm³
    4. 84.78 cm³
    Show answer

    Answer: 113.04 cm³

    V = (4/3)πr³ = (4/3) × 3.14 × 27 = (4/3) × 84.78 = 113.04 cm³.

  5. Volume is always expressed in _____ units.

    Show answer

    Answer: cubic

    Volume measures three-dimensional space, so units are always cubed (e.g., cm³).

Get a personalized course on Volume of 3D Figures built around your child.